A345056 -id:A345056 - cp's OEIS frontend

A345057 Distance to the next larger number k > n for which A011772(k) = A011772(n), or 0 if no such number exists.

0, 4, 0, 10, 5, 0, 14, 12, 3, 0, 44, 6, 13, 14, 0, 46, 17, 18, 38, 4, 0, 11, 230, 6, 25, 13, 90, 0, 29, 10, 62, 40, 33, 34, 70, 0, 37, 57, 39, 20, 41, 28, 86, 4, 0, 23, 1034, 40, 7, 25, 102, 78, 53, 9, 0, 28, 114, 145, 1652, 60, 61, 62, 63, 190, 260, 0, 134, 68, 23, 140, 284, 24, 73, 37, 25, 38, 154, 0

Offset: 1

Antti Karttunen, Jun 07 2021

nonn

Cf. A000217 (positions of zeros), A002024, A011772, A066561, A345056, A345058.

A011772(n) = { if(n==1, return(1)); my(f=factor(if(n%2, n, 2*n)), step=vecmax(vector(#f~, i, f[i, 1]^f[i, 2]))); forstep(m=step, 2*n, step, if(m*(m-1)/2%n==0, return(m-1)); if(m*(m+1)/2%n==0, return(m))); }; \\ From A011772
A345057(n) = { my(x=A011772(n), y=binomial(x+1,2)); for(i=1+n,y,if(A011772(i)==x,return(i-n))); (0); };

a(n) = A345056(n) - n.

A345058 Number of distinct k > 0 for which A011772(k) = n.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 2, 7, 4, 2, 2, 4, 4, 3, 1, 5, 4, 4, 3, 7, 2, 3, 2, 7, 6, 4, 3, 3, 4, 7, 4, 13, 3, 4, 2, 4, 6, 3, 4, 7, 6, 3, 4, 7, 1, 5, 2, 7, 11, 6, 2, 7, 4, 5, 2, 11, 3, 4, 2, 7, 8, 5, 5, 7, 2, 3, 4, 4, 5, 3, 4, 13, 9, 6, 6, 9, 2, 4, 4, 8, 15, 6, 2, 11, 3, 3, 3, 7, 6, 3, 1, 5, 4, 7, 3, 13, 10, 11, 7, 6, 6, 5, 4, 8, 2

Offset: 1

Antti Karttunen, Jun 07 2021

nonn

Question: Are there any numbers n other than 1, 3, 15, 45, 91 for which a(n) = 1?

			A011772 obtains the value 6 only as A011772(7)=6 and A011772(36)=6, therefore a(6) = 2.

Cf. A000217, A002024, A011772, A066561, A345056, A345057.

A011772(n) = { if(n==1, return(1)); my(f=factor(if(n%2, n, 2*n)), step=vecmax(vector(#f~, i, f[i, 1]^f[i, 2]))); forstep(m=step, 2*n, step, if(m*(m-1)/2%n==0, return(m-1)); if(m*(m+1)/2%n==0, return(m))); }; \\ From A011772
A345058(n) = { my(x=A011772(n), y=binomial(x+1,2)); sum(i=1,y,(A011772(i)==x)); };

a(n) = Sum_{i=1..A000217(n)} [A011772(i) = A011772(n)], where [ ] is the Iverson bracket.

cp's OEIS Frontend

A345057 Distance to the next larger number k > n for which A011772(k) = A011772(n), or 0 if no such number exists.

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